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I figured if I proved that $f'(x)=0$ then that would be it because, if first derivative is zero, than the function has constant value for every point of its domain. But I don't know how to prove that $f'(x)=0$ using given condition about limits. I thought about $\lim_{x\to\infty}(f(x)+f'(x))=\lim_{x\to\infty}f(x)+\lim_{x\to\infty}f'(x)$ but I failed to find out why is $\lim_{x\to\infty}f'(x)$ equal to zero.

Bernard
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Nigruteen
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1 Answers1

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Here is just a thought, so correct me if I'm wrong:

We can have 3 cases :

(i) $\lim_{x \to \infty} f'(x)>0$

Here it will imply the function is increasing even as we approach $\infty$, so as $x$ goes on increasing to a very large number, $f(x)$ will blow up to $\infty$

Thus, $\lim_{x \to \infty} f(x)+f'(x) = (something \ blowing \ to \ \infty) + (some \ positive) \neq c$

This can definitely not reach a constant $c \in \mathbb{R}$.

(ii) $\lim_{x \to \infty} f'(x)<0$

Similar to case (i), here $f(x)$ will blow down to $-\infty.$ Thus, $\lim_{x \to \infty} f(x) + f'(x) = (-\infty) + (some \ negative) \neq c \in \mathbb{R}$

(iii) The only case we are left with is $\lim_{x \to \infty} f'(x)=0$

We can see two sun-cases here:

(a) $$\lim_{x \to \infty} f(x)= c$$

checking this with the question it satisfies. So this is a valid answer.

(b) $$\lim_{x \to \infty} f(x)= \pm \infty$$

This would again not satisfy the condition of $\lim_{x \to \infty} f(x) +f'(x) = c$

So, I believe every possible case is accounted for, and only one case is satisfying the conditions, so

For $$\lim_{x \to \infty } f(x) + f'(x) = c \in \mathbb{R}$$

It is necessary that $\lim_{x \to \infty} f'(x) =0$ and $\lim_{x \to \infty}f(x)=c$

Saket Gurjar
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  • Can't we have that the limit of the derivative does not exist? – Botond May 25 '20 at 19:37
  • The function is differentiable @Botond – Saket Gurjar May 25 '20 at 19:37
  • Yes. So is the $\sin$ function, but the limit of it's derivative does not exist. – Botond May 25 '20 at 19:39
  • @Botond if $\lim_{x \to \infty} f'(x)$ does not exist, then how can $\lim_{x \to \infty} f(x) +\lim_{x \to \infty} f'(x)$ be a definite constant? – Saket Gurjar May 25 '20 at 19:42
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    I would say this proof is valid. – Nigruteen May 25 '20 at 19:43
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    If the limit of $g$ does not exist, we can still have that the limit of $f+g$ exists, but we can't separate the limit of the sum into a sum of limits. For example, take $f=\sin$, $g=-\sin$. But I'll think a little bit and see if I can come up with an example to my question. – Botond May 25 '20 at 19:45
  • @Botond I think I am getting your point, but then you'll have to either find a counter example or prove that there is no such $f(x)$. I'd say it has to be the latter because this seems to be a rather common question (see the reason for the close vote on the question), and such a flaw would have come up before. – Saket Gurjar May 25 '20 at 19:51
  • But again will have to come up with a mathematical argument to prove so. – Saket Gurjar May 25 '20 at 19:52